A closed-form solution for paraperspective reconstruction

We address the problem of 3D reconstruction from image features tracked along a sequence. The most precise algorithms compute the maximum likelihood (ML) estimate and are iterative. They need an approximate 3D reconstruction as starting position. For that purpose, we propose a closed-form expression of paraperspective reconstruction. A matrix that approximately verifies the properties of a paraperspective projection matrix is first built, as in Christy and Horaud (1994) or Poelman and Kanade (1997). Our contribution lies in showing how to transform this matrix so that it exactly verifies the properties of paraperspective projection matrices. This is done by a closed form expression, in which the depth of the camera is also retrieved. The camera position is then found directly, instead of being obtained as the solution of a non-linear optimization problem, like in Poelman and Kanade (1997)